Double parton distributions in Light-Front constituent quark models
Matteo Rinaldi, Sergio Scopetta, Marco Traini, Vicente Vento
aa r X i v : . [ h e p - ph ] N ov Few-Body Systems manuscript No. (will be inserted by the editor)
Matteo Rinaldi · Sergio Scopetta · Marco Traini · Vicente Vento
Double parton distributions in Light-Frontconstituent quark models
Received: date / Accepted: date
Abstract
Double parton distribution functions (dPDF), accessible in high energy proton-proton andproton nucleus collisions, encode information on how partons inside a proton are correlated among eachother and could represent a tool to explore the 3D proton structure. In recent papers, double partoncorrelations have been studied in the valence quark region, by means of constituent quark models.This framework allows to understand clearly the dynamical origin of the correlations and to establishwhich, among the features of the results, are model independent. Recent relevant results, obtainedin a relativistic light-front scheme, able to overcome some drawbacks of previous calculations, suchas the poor support, will be presented. Peculiar transverse momentum correlations, generated by thecorrect treatment of the boosts, are obtained. The role of spin correlations will be also shown. In thiscovariant approach, the symmetries of the dPDFs are unambiguously reproduced. The study of theQCD evolution of the model results has been performed in the valence sector, showing that, in somecases, the effect of evolution does not cancel that of correlations.
Keywords proton structure · relativistic models In high energy hadron-hadron collisions, more than one parton in each of the hadron can contributeto the cross section. This is the multiple partonic interactions (MPI) phenomenon which, even if itscontribution is suppressed by a power of Λ QCD /Q with respect to the single parton interaction, with Q the center-mass energy in the collision, has been already observed (see, e.g. , Ref. [1]). In this scenarioMPI represent a background for the search of new Physics, e.g. , at the LHC. In this work we focusour attention to the double parton scattering (DPS) which can be observed in many channels, e.g. , W W with dilepton productions and double Drell-Yan processes (see, e.g.
Refs. [2; 3; 4; 5] for recentreviews). At the LHC, DPS, whose evidence has been observed [6], represents also a background for
M. RinaldiDipartimento di Fisica e Geologia, Universit`a degli Studi di Perugia, and INFN, sezione di Perugia, 06100Perugia, ItalyE-mail: [email protected]. +39 075 5852793S. ScopettaDipartimento di Fisica e Geologia, Universit`a degli Studi di Perugia, and INFN, sezione di Perugia, 06100Perugia, ItalyM. TrainiDipartimento di Fisica, Universit`a degli studi di Trento, and INFN - TIFPA, 38123 Trento, ItalyV. VentoDepartament de Fisica Te`orica, Universitat de Val`encia and Institut de Fisica Corpuscular, Consejo Superiorde Investigaciones Cient´ıficas, 46100 Burjassot (Val`encia), Spain
Higgs production. In this framework the DPS cross section can be written, following [7], in terms ofthe so called double parton distribution functions (dPDFs), F ij ( x , x , z ⊥ , µ ), which describe the jointprobability of finding two partons of flavors i, j = q, ¯ q, g with longitudinal momentum fractions x , x and transverse separation z ⊥ inside the hadron: dσ = 1 S X i,j,k,l Z d z ⊥ F ij ( x , x , z ⊥ , µ ) F kl ( x , x , z ⊥ , µ )ˆ σ ik ( x x √ s, µ )ˆ σ jl ( x x √ s, µ ) . (1)The partonic cross sections ˆ σ refer to the hard, short-distance processes, S is a symmetry factor,present if identical particles appear in the final state and µ is the renormalization scale which istaken, for simplicity, to be the same for both partons. For the evaluation of the DPS contributions toproton-proton scattering at LHC kinematics, the following approximation, for the dPDF, is usuallymade: F ij ( x , x , z ⊥ , µ ) = q i ( x , µ ) q j ( x , µ ) θ (1 − x − x )(1 − x − x ) n T ( z ⊥ , µ ) , (2) i.e. , a complete factorized form of the dPDF is assumed. In particular the z ⊥ and x − x dependencesare factorized and the standard single parton distribution functions (PDF), q ( x ), are introduced. Thismeans that possible double parton correlations between the two interacting partons are neglected.dPDFs are non perturbative quantities so that they cannot be easily evaluated in QCD. As it happensfor the PDF, a useful procedure for their estimate is a calculation at the hadronic scale, Q ∼ Λ QCD , bymeans of quark models. In order to compare the obtained results with data taken, e.g. , at high energyscales,
Q > Q , it is necessary a perturbative QCD (pQCD) evolution of the model calculations, usingthe dPDFs evolution equations known since a long time ago [8; 9]. By using this procedure, future dataanalysis of the DPS processes could be guided, in principle, by model calculations. The first modelevaluations of the dPDF have been the ones in Refs.[10; 11]. In the first scenario use has been madeof a modified version of the MIT bag model in the cavity-approximation in order to introduce doubleparton correlations by hand, recovering momentum conservation. In the second case the dPDFs havebeen calculated in a non relativistic (NR) constituent quark model (CQM) framework, since CQM,in the valence region, predict PDFs, generalized parton distribution functions (GPDs) and transversemomentum dependent parton distributions (TMDs) rather well (see, e.g. , Refs. [12; 13; 14]). Theseexpectations motivated the analysis of Refs. [11; 15] and the present one. The main results found inRefs. [10; 11] are that, in the valence quark region, the approximations used to write Eq. (2) are badlyviolated. In the CQM picture, where the dynamical origin of double correlations is clear, the origin ofthis violation can be properly understood. One should notice that both the analyses of Refs. [10; 11]have some inconsistencies. First, dPDF do not vanish in the non physical region, x + x > i.e. , theyhave a wrong support. Moreover, as already pointed out, in order to obtain some information on thedPDF at small values of x and at high Q , where LHC data are taken, the pQCD evolution of thecalculated dPDF is necessary. In a recent paper of ours [15], a CQM calculation of the dPDF has beenperformed including relativity through a fully Poincar´e covariant Light-Front (LF) approach. Thanksto this treatment it is possible to study strong interacting systems with a fixed number of on-shellconstituents (see Refs. [16; 17] for general reviews). Moreover, being the hyperplane of the LF, i.e. , theplane where the initial conditions are defined, tangent to the Light-Cone, the Deep Inelastic Scattering(DIS) phenomenology is automatically included into the scheme. In particular, in this framework, whichhas been used extensively for hadronic calculations, (see, e.g. , Refs. [18; 19; 20], some symmetries of thedPDF are restored and the bad support problem is fixed, so that the pQCD evolution of the dDPDFsis more precise. The results of this analysis will be summarized in the following sections. In this section the procedure adopted for the LF calculation of the dPDFs will be presented. Thevalidity of the approximations Eq. (2) in this relativistic scenario will be checked. To this aim, the LFapproach has been chosen due to its nice properties, in particular the fact that LF boosts and pluscomponent of the momenta ( a + = a + a z ) are kinematical operators. The Fourier- transform of thedPDF F λ ,λ ij ( x , x , k ⊥ ) = Z d z ⊥ e i z ⊥ · k ⊥ F λ ,λ ij ( x , x , z ⊥ ) , (3) will be analized. In the above equation, the dPDF is introduced for two quarks of flavors i and j andhelicities λ i ( j ) , respectively. The full procedure of the calculations of the dPDF in the LF approach,starting from the formal definition of this quantity, and thanks to a proper extension of the proceedingpresented in Ref. [20; 21], developed in that case for the GPDs calculations, is shown in details inRef. [15]. Here the main steps are summarized. In particular, one can start from the expression of alight-cone correlator (see, e.g. , Ref. [4]), written in terms of the proton state and of LF quantized fieldsof the interacting quarks. In order to find a general expression of the dPDF in the valence region, usehas been made of the so called “LF wave function” (LFWF) representation [17] to describe the protonstate. In this formalism the latter quantity is written as a sum over partonic Fock states with all thecorrect normalizations preserved. In the LFWF representation, only the first term of this summation,representing the contribution of the valence quarks, has been taken into account. A crucial point of theprocedure is the possibility of describing the LF proton state starting from the Instant Form (canonical)one, where most quark models are developed. To this aim the following relation between one particleLF state, | k , λ i [ l ] , and the corresponding IF one, | k , λ ′ i [ i ] , has been used: | k , λ i [ l ] = (2 π ) / p m + k X λλ ′ D / λλ ′ ( R cf ( k )) | k , λ ′ i [ i ] , (4)where the Melosh rotation, which allows to rotate the canonical helicity, λ , into the LF spin, λ ′ , isintroduced: D / µλ ( R cf ( k )) = h µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m + x i M − i σ i · (ˆ z × k ⊥ ) p ( m + x i M ) + k ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i . (5)In the above equation, x i = k + i P + is the longitudinal momentum fraction carried by the i parton,with P + the plus component of the proton momentum, M = P i p m + k i the total free energy massof the partonic system, k iz = − ( m + k i ⊥ − k +2 i ) / (2 k + i ) , µ and λ generic canonical spins. Actually, weare interested in the proton, a composite system, and the validity of Eq. (4), supposed for free states,is questionable. Nevertheless, in the following, we will use a relativistic mass equation built in accordwith the Bakamjian-Thomas Construction of the Poincar´e generators, and Eq. (4) can be used (seeRef. [16]). Using a lengthy but straightforward procedure, a final expression of the dPDF is obtained.It reads [15]: F λ ,λ q q ( x , x , k ⊥ ) = 3( √ Z Y i =1 d k i X λ fi τ i δ X i =1 k i ! Ψ ∗ (cid:18) k + k ⊥ , k − k ⊥ , k ; { λ fi , τ i } (cid:19) (6) × b P q (1) b P q (2) b P λ (1) b P λ (2) Ψ (cid:18) k − k ⊥ , k + k ⊥ , k ; { λ fi , τ i } (cid:19) × δ (cid:18) x − k +1 P + (cid:19) δ (cid:18) x − k +2 P + (cid:19) . The canonical proton wave function ψ [ c ] is embedded in the function Ψ here above, which can bewritten as follows: Ψ ( k , k , k ; { λ fi , τ i } ) = Y i =1 X λ ci D ∗ / λ ci λ fi ( R cf ( k i )) ψ [ c ] ( { k i , λ ci , τ i } ) , (7)where here λ ci , τ i are the canonical partonic helicity and isospin respectively and the short notation { α i } instead of α , α , α is introduced. Moreover isospin and spin projection operators are introducedin order to access dynamical correlations for the unpolarized and longitudinal polarized i quark of agiven flavor: ˆ P u ( d ) ( i ) = 1 ± τ ( i )2 , ˆ P λ k ( i ) = 1 + λ k σ ( i )2 . (8) uu ( x , . , k ⊥ ) / uu ( . , . , k ⊥ ) x | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 1 . Fig. 1
The ratio r , Eq. (11), for five values of k ⊥ . ∆ u ∆ u ( x , . , k ⊥ ) / ∆ u ∆ u ( . , . , k ⊥ ) − − x | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 0 . | k ⊥ | = 1 . Fig. 2
The ratio r , Eq. (11), for five values of k ⊥ . A crucial point of this analysis, as already pointed out, is that now the plus components of themomenta are totally kinematical so that one finds: P + = X i k + i = M , (9)in the intrinsic frame where k + k + k = 0 . Thanks to the relation Eq. (9) the delta function,defining the longitudinal momentum fraction carried by the parton in Eq. (6), can be properly solvedwithout any additional approximation, at variance with what happens in the instant form calculationin Ref. [11]. As a direct consequence, the bad support problem does not show up. The followingdistributions, different from zero for an unpolarized proton, have been calculated: uu ( x , x , k ⊥ ) = X i,j = ↑ , ↓ u i u j ( x , x , k ⊥ ) , ∆u∆u ( x , x , k ⊥ ) = X i,j = ↑ , ↓ ( − i + j +1 u i u j ( x , x , k ⊥ ) (10)In order to calculate now the dPDFs, in particular to check whether the approximation, Eq. (2),holds, a proper CQM has to be chosen. To this aim, in order to have a fully consistent procedure, arelativistic model has been used, in particular the one described in Ref. [19], a hyper-central CQM.This model provides a reasonable description of the light hadronic spectrum, it has been used for theestimate of PDFs and GPDs in Refs. [19; 20; 21; 22; 23] and, for the present analysis, since no data areavailable for the dPDF, it can be used as laboratory to predict the most relevant features of dPDF. Inparticular, in Figs. 1 and 2 the following ratios have been shown for five values of k ⊥ : r = uu ( x , . , k ⊥ ) uu (0 . , . , k ⊥ ) , r = ∆u∆u ( x , . , k ⊥ ) ∆u∆u (0 . , . , k ⊥ ) . (11)In particular, the value x = 0 . r and r depend on k ⊥ so that a factorized formof the dPDFs for the k ⊥ dependence is not supported by this approach. One can easily realize, in fact,that a factorized expression for the dPDF would yield k ⊥ -independent r and r . It is also importantto notice that the amount of the violation is directly related to the Melosh rotation contributions, amodel independent relativistic effect. In Fig. 3 the following ratios r = 2 uu ( x , x , k ⊥ = 0) u ( x ) u ( x ) , r = C∆u∆u ( x , x , k ⊥ = 0) ∆u ( x ) ∆u ( x ) , (12)where: C = [ R dx∆u ( x )] R dx dx ∆u∆u ( x , x , k ⊥ = 0) , (13) Fig. 3 a) The ratio r , Eq. (12), at the hadronic scale; b) the same quantity at a scale Q = 10 GeV ; c) theratio r , Eq. (12), at the hadronic scale µ ; d) this last quantity at a scale Q = 10 GeV . The vertical scaleof panels (b) and (d) is reduced by a factor of 2 with respect to panels (a) and (c), respectively. involving the standard PDFs in the denominators, calculated within the same CQM, have been pre-sented. In Figs. 3(a), 3(c) these quantities are shown at the hadronic scale obtaining that a factorizedapproximation of the dPDF, in terms of the single PDFs, is not favored in the valence quark region.Notice that the factors 2 and C , in Eq. (12), are inserted in order to have these ratios equal to 1 inthe kinematical region where the approximation, Eq. (2), is valid. Moreover, in Ref. [15] the pQCDevolution at Leading-Order of the calculated dPDFs has been also presented. In particular, for themoment being, the evolution of the dPDF has been addressed for k ⊥ = 0, taking the same scale forthe two acting partons and analyzing only the valence quark contributions. In this case the evolutionequations are obtained as a direct generalization of the well known DGLAP ones, see Refs. [8; 9].Since only the non singlet case has been studied here, one needs to solve the homogeneous part ofthe generalized DGLAP equations by using the Mellin transformations of the dPDFs, see Ref. [15] fordetails. If we use their simple ansatz as an input to our calculation, our code reproduce the resultsof the authors of Ref. [24]. Once the dPDFs have been evaluated at a generic high energy scale, e.g. , Q = 10 GeV , the ratios r and r have been shown again in Figs. 3(b), 3(d). The most importantresults of this analysis are that, for small values of x , where data are taken, e.g. , at the LHC, r ∼ r , in Fig 3(d), it is found that double spin correlations still contribute. In this work, dPDFs contributing to the DPS cross section have been calculated by means of aLF CQM. The main achievement is the fully Poincar´e covariance of the description, which allows torestore the expected symmetries, and the vanishing of the dPDF in the forbidden kinematical region, x + x >
1. In the analysis of the dPDFs at the hadronic scale we found that the approximationsof these quantities with a complete factorized ansatz, in the x − x and ( x , x ) − k ⊥ dependences,is violated, in agreement with previous results [10; 11]. Moreover, a pQCD analysis of the dPDFs,necessary to evaluate these quantities at higher energy scales with respect to the hadronic one wherethe CQM predictions are valid, has been also performed. For the unpolarized quarks case the dynamicalcorrelations are suppressed in the small x region, while double spin correlations are found to be stillimportant. Further analysis, including into the scheme non perturbative sea quarks and gluons andthe evolution of the singlet sector, important to describe the dPDF at low x , are in progress, as wellas the study of correlations in pA scattering, along the line of Ref. [25]. References
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